The subject of Rotatory Motion continued.--A Ball, by having a peculiar spinning motion imparted to it, may be made to stop short, or to retrograde, though it meets not with any apparent obstacle.--The rectilinear path of a Spherical Body influenced by its rotatory motion.--Bilboquet, or Cup and Ball.--The joint forces which enable the Balancer to throw up and catch his Balls on the full gallop.--The Hoop.--The Centre of Percussion.--The Whip and Peg-top.--Historical Notices.--The power by which the Top is enabled to sustain its vertical position during the act of spinning.--The sleeping of the Top explained.--The force which enables it to rise from an oblique into a vertical position.--Its gyration.

“Tom, do you remember that I told you a few days ago,” said Mr. Seymour, “that, by giving a revolving body a peculiar spinning motion, certain effects were produced, which I should, on some future occasion, take into consideration?”

“To be sure I do,” replied Tom.

“Well, then, attend to me.”

Mr. Seymour took a marble, and, placing it on the ground, gave it an impulse forward by pressing his forefinger upon it: the marble darted forward a few paces, after which it rolled back again.

“That is most extraordinary!” cried Tom; “the marble came back to your hand, as it were, of its own accord, and without having met with any obstacle.”

“And you, no doubt,” said Mr. Seymour, “regard it as contrary to the well known law, that a body once put in motion, in any direction, will continue to move in that direction until some foreign cause oppose it.”

“It really would appear so.”

“It is, however, far otherwise; the force which I imparted to the marble communicated to it two kinds of motion; the one projecting it forward, the other producing a rotatory motion round its axis, in a direction opposite to that of its rectilinear course; and the consequence was simply this, that when the former motion, on account of the friction of the marble on the ground, was destroyed, the rotatory motion continued, and by thus establishing an action in an opposite direction, caused the marble to retrograde. If, however, you will fetch your hoop, I will demonstrate the fact on a larger scale.”

Tom accordingly produced the hoop; and Mr. Seymour projected it forward, giving to it, at the same instant, a spinning motion in an opposite direction. The hoop proceeded forward to a certain distance, when it stopped, and then ran back to the hand.

“Let me beg you,” said Mr. Seymour, “to treasure this fact in your memory; you perceive by it how greatly the progressive direction of a body may be influenced by a rotatory motion around its axis; and, indeed, the theory of the rifle gun(20) is easily deduced from it. It will also explain the effect which a rotatory motion produces in steadying or disturbing the direction of a projectile. It is for such a reason that the balancer constantly whirls round his balls or oranges, as he throws them into the air, with the intention of catching them again; and that in playing at Bilboquet, or cup and ball, you find it necessary to give a spinning motion to the ball, in order to catch it on the spike--but we will consider this subject presently. I am now desirous of laying down a few propositions upon the subject of rotation, the knowledge of which is essential for the explanation of the motions of revolving bodies.”

Mr. Seymour proceeded to state that every body had three principal axes upon which it might revolve, but that the shortest was the only one upon which it could permanently and steadily rotate--that should it, in consequence of the impulse given to it, begin to spin upon any other than the shortest axis, it would, during its revolutions, be constantly showing a tendency to approach it; whence it followed that, under such circumstances, it would be unsteady and wabbling in its motions.

In order, however, to make this proposition intelligible to the children, Mr. Seymour performed the following simple experiment.

Having tied some string to a common curtain ring, as represented by figure 1, he twisted it round by means of his thumb and finger, until it acquired considerable velocity, when the ring was seen to rise gradually into the position represented by fig. 2. Thus, in the simplest manner, was a revolving body shown to exchange its longer for its shorter axis.

The children declared that they perfectly comprehended the subject, and Tom observed that, in future, whenever he wished to make a ball spin steadily, he should take care to make it turn on its shortest axis.

“You are quite right, Tom,” said Mr. Seymour; “and the skilful bowler at cricket, in order to give his ball a steady axis of rotation, always holds it with the seam across, so that the tips of his fingers may touch, and he takes care to hold it only with such a grasp as may be sufficient to steady it, for by a turn even of the wrist it may be made to proceed unsteadily; and this leads me to consider another equally important proposition--viz. that the axis of rotation should coincide with the direction in which it is moving forward, or, in other words, with its line of motion. Now, where this is not the case, it is evident that the unequal action of the air will cause the body to deviate from its straight course, since its two sides, having different velocities (the rotatory and progressive motions conspiring on one side, while they are in opposition on the other), will be differently affected by such resistance; the resistance, of course, increasing with the velocity. It is upon this principle,” continued Mr. Seymour, “that Sir Isaac Newton has explained the irregular motion of the tennis-ball.”

“But do explain to us, papa,” said Louisa, “why it is so necessary to spin the ball in order to catch it on the spike?”

“Rotatory motion, my dear, when directed according to the principles I have endeavoured to enforce, will always steady the course of a body. In playing at bilboquet, your object is so to throw up the ball that its hole may descend perpendicularly upon the spike which is held for its reception; and in order to accomplish this, you make the ball spin upon an axis, at the extremity of which is the hole; the consequence is obvious.”

Louisa observed, that she well remembered an allusion to this game in Miss Edgeworth’s Essays on Education; and that, unless she was much deceived, the advantage to be gained by spinning the ball was referred to centrifugal force, and its effect in preserving the “parallelism of motion.”

“I do not recollect the passage,” answered her father, “but I will admit that the centrifugal force is indirectly instrumental to the effect, although, in my view of the subject, it is more philosophical to refer it at once to the creation of an appropriate axis of rotation.”

“I well remember,” observed Tom, “that the rider at Astley’s whirled round the oranges as he threw them into the air.”

“And I hope that you are now not only acquainted with the principle which rendered such a rotatory motion necessary, but that which must make the shorter the more eligible axis for effecting his purpose;--but can you tell me how it could have happened, that the oranges, which were thrown perpendicularly upwards while the horseman was on the full gallop, should have fallen again into his hand?”

“Ay,” said Louisa, “that puzzled me exceedingly; I should have thought he would have ridden away from them, and that they must have fallen several feet behind him.”

“What say you, Tom, to that?” enquired Mr. Seymour.

“I suppose,” replied Tom, “that the rider calculated upon the distance he would pass forward before they could fall, and projected them accordingly.”

“No, indeed; there is no calculation in the case, nor is any art used to throw the oranges in advance: they are projected perpendicularly from the hand; and if you will only recall to your mind the subject of the ‘Composition of Forces,’ the mystery will vanish.”

“I see it all clearly,” cried Tom; “the orange partakes of the progressive motion of the rider; when, therefore, he throws it upwards, it is influenced by two forces which are in the direction of the two sides of a parallelogram, and it consequently describes the diagonal.”

“You are quite right; but you doubtless will perceive that, instead of a straight line, the orange will describe a parabolic curve.”

“For the same reason, I suppose,” said Tom, “that the stone from the sling described a curve?”

“Certainly; but see, I have a diagram which will explain the subject more clearly.”

“The orange, as it is thrown into the air, is influenced by two forces: the one arising from the progressive motion of the rider, the other from the projectile force imparted to it. These two forces are in the direction of the adjacent sides of a parallelogram, and were it not for the operation of gravity, the body would accordingly describe its diagonal in the same space of time as it would have described one of the sides.^[18] The influence of gravity, however, not only deflects it from a right line into a curve, but diminishes its force, so that instead of arriving at the opposite angle of the parallelogram a, its greatest altitude will be short of that point; it will then descend through a similar curve; and, since the times of ascent and descent are equal,^[19] it will reach the hand of the rider at the very moment he is prepared to receive it; for the orange will have traversed the parabolic curve in the same space of time as the horseman required for passing from one extremity of the curve to the other.”

Mr. Seymour having concluded this explanation, much to the satisfaction of the young party, observed that the present occasion was an appropriate one for the introduction of some remarks on the favourite pastime of the Hoop.

“It is a classical pastime,” exclaimed the vicar, “and was as common with the Greeks and Romans as it is with boys of the present generation.”

“And it has the advantage,” added Mr. Seymour, “of sending the tide of life in healthful currents through the veins.”

Tom began to trundle his hoop along the gravel walk.

“Stop, stop, my dear boy,” cried his father, “you seem to have forgotten our compact, that every toy should be fairly won before it was played with. Come upon the lawn, and let me ask you some questions relative to the motions of the hoop. Can you make it stand still upon its edge?”

“Not readily,” was Tom’s reply.

“And yet,” continued Mr. Seymour, “during its progressive motion, it rolls on its edge without any disposition to fall: how happens that?”

“It is owing to the centrifugal force, which gives it a motion in the direction of a tangent to the circle, and, consequently, overcomes the force of gravity.”

“Your answer is pat,” replied his father: “as long as you give your hoop a certain degree of velocity, the tangential, or centrifugal force, overcomes gravity, in the manner you have already witnessed;^[20] but, when that is slackened, the hoop will fall on its side; not, however, until it has made several complete revolutions. Now, answer me another question. Why is it so difficult to make the hoop proceed straight forward, without turning to the right or left?”

“I suppose it arises from the same cause as that which altered the direction of my marble as it ran along--the inequality of the ground.”

“That,” replied his father, “would undoubtedly have its influence; but it is principally to be referred to the impossibility of your constantly giving a straight blow by the stick. When it is moving forward, a slight inclination towards either side will cause the parts to acquire a motion towards that side, those which are uppermost being most affected by it; and this lateral, or sideway motion, assisted sometimes by the irregular curvature of the hoop, causes its path to deviate from a rectilinear direction; so that, instead of moving straight forward, it turns to that side towards which it began to incline; and, in this position, its tendency to fall is still farther counteracted by the centrifugal force. It is from a similar cause that the bullet, unless rifled, will have a tendency to go to the right or left, from any unequal impulse which it may have received at the moment of its exit from the barrel. I have yet one other question, and, as its answer will lead us into the consideration of a mechanical subject of some importance, I must beg you to bestow all your attention. In trundling your hoop, have you not often observed that, although the blow inflicted upon it by your stick might have been violent, yet the effect produced by it was comparatively small, in consequence of the hoop having been struck by a disadvantageous part of the stick?”

“Certainly! I have frequently observed that, if the hoop is struck by the stick either too near the hand, or too near the end, much of its force is lost; and I have also noticed the same thing in striking the ball with my cricket-bat.”

“The fact is,” said Mr. Seymour, “that every striking body has what is termed its centre of percussion, in which all the percutient force of a body is, as it were, collected; thus, a stick of a cylindrical figure, supposing the centre of motion at the hand, will strike the greatest blow at a point about two-thirds of its length from the wrist. I may, perhaps, at some future time, return to this subject, and explain several mechanical effects which are dependent upon it.(21) Now away with you, and trundle your hoop, or spin your top; as soon as the vicar arrives I will rejoin you.”

In the course of an hour Mr. Seymour and his reverend friend proceeded to the play-ground, where they found the children busily engaged in their several diversions.

“I rejoice to find you at so classical a pastime,” said the vicar, as he approached Tom, who was busily engaged in spinning his top. “The top, my boy, is a subject which the great Mantuan bard did not consider beneath the patronage of his muse: but, hey-day! this is not the ‘volitans sub verbere turbo’ of the immortal Virgil; the top of antiquity was the whip-top, the peg-top is a barbarous innovation of modern times: a practical proof of the degeneracy of the race. Even boys, forsooth, must now-a-days have their activity cramped by inventions to supersede labour: well may we regard the weapons, which our sturdy ancestors wielded as instruments rather calculated for giants than men, if such pains be taken to instil into the minds of youth the mischievous spirit of idleness.”

“My dear sir,” said Tom, who was always grieved at displeasing the vicar, “if it will gratify you, I will spin my whip-top, for I have an excellent one which my papa has lately given me.”

“Well said! my dear boy. ‘Puer bonÆ spei.’--What a pity would it be to damp so noble a spirit; get your whip-top.”

Tom accordingly placed the Virgilian top upon the ground, and as the boy plied the whip, so did the vicar lash the air with his quotation; running round the top in apparent ecstasy, while he repeated the well-known lines from the seventh Æneid:--

As Mr. Twaddleton thus gave vent to that fervour which was ever kindled by collision with Virgil, Tom gave motion to his top, which swaggered about with such an air of self-importance, that, to the eye of fancy, it might have appeared as if proudly conscious of the encomiums that had been so liberally lavished upon it.

“The Grecian boys, as Suidas informs us, played also with this top,” continued the vicar.

“And pray, may I ask,” said Mr. Seymour, “whether it was not introduced into this country by the Romans?”

“Probably,” replied the vicar. “Figures representing boys in the act of whipping their tops first appear in the marginal paintings of the manuscripts of the fourteenth century; at which period, the form of the toy was the same as it is at present, and the manner of impelling it by the whip can admit of but little if any difference. In a manuscript,^[22] at the British Museum, I have read a very curious anecdote which refers to Prince Henry, the eldest son of James the First; with your permission I will relate it to you.”

Here the vicar extracted a memorandum-book from his pocket, and read the following note:--

“The first tyme that he, the prince, went to the towne of Sterling to meete the king, seeing a little without the gate of the towne a stack of corne, in proportion not unlike to a topp, wherewith he used to play, he said to some that were with him, ‘Loe there is a goodly topp:’ whereupon one of them saying, ‘Why doe you not play with it then?’ he answered, ‘Set you it up for me, and I will play with it.’”

“Was not that a clever retort of the young prince?” said the vicar, as he returned the manuscript into his memorandum-book; “and I think it must have confounded the courtier who could have asked so silly a question.”

“Well, Tom,” said Mr. Seymour, “let us see whether you can set up your own top, so that it shall stand steadily on its point.”

“I have often tried that experiment,” answered Tom, “but could never succeed in keeping the line of direction within its narrow base.”

“And yet, when in rotatory motion, its erect position is maintained without difficulty; how is that?”

“Is it not owing to the centrifugal force?” asked Tom.

“Undoubtedly: but as the subject is highly interesting, I will endeavour to explain it more fully. You must, however, first obtain permission from the vicar to spin your humming-top, for that will better illustrate the phenomena which it is my wish to examine.”

“If your object is the exercise of the body, let us spin the whip-top,” replied the vicar; “but if you wish to exercise the boy’s mind, I cannot object to your selecting the top best calculated to fulfil that desire.”

Tom, having accordingly prepared his top, pulled the string, and set the wooden machine spinning on the floor.

“Now, Tom, I will explain to you the reason of the top being able to sustain its vertical position. You have already learned, from the action of the sling, that a body cannot move in a circular path, without making an effort to fly off in a right line from the centre;^[23] so that, if a body be affixed to a string, and whirled round by the hand, it will stretch it, and in a greater degree according as the circular motion is more rapid.”

“Certainly,” said Tom.

“The top, then, being in motion, all its parts tend to recede from the axis, and with greater force the more rapidly it revolves: hence it follows that these parts are like so many powers acting in a direction perpendicular to the axis; but, as they are all equal, and as they pass all round with rapidity by the rotation, the result must be that the top is in equilibrio on its point of support, or on the extremity of the axis on which it turns. But see, your top is down.”

“And what is the reason,” asked Tom, “of its motion being stopped?”

“I can answer that question, papa,” said Louisa; “is it not owing to the friction of the ground?”

“Certainly; that has, doubtless, its influence: but the resistance of the air is also a powerful force upon this occasion. A top has been made to spin in vacuo as long as two hours and sixteen minutes.^[24] But come, Tom, spin your top once more. Observe,” exclaimed Mr. Seymour, “how obliquely the top is spinning. It is now gradually rising out of an oblique position;--now it is steadily spinning on a vertical axis;--and now its motion is so steady, that it scarcely seems to move.”

“It is sleeping, as we call it,” said Tom.

“Its centre of gravity is now situated perpendicularly over its point of support, which is the extremity of the axis of rotation: but attend to me,” continued Mr. Seymour, “for I am about to attempt the explanation of a phenomenon which has puzzled many older and wiser philosophers than yourselves. It is evident that the top, in rising from an oblique to a vertical position, must have its centre of gravity raised; what can have been the force which effected this change?”

“Was it the centrifugal force?” asked Tom.

“Certainly not,” said Mr. Seymour, “as I will presently convince you.”

“Then it must have been the resistance of the air,” said Louisa.

“No; nor was it the resistance of the air,” replied her father: “for the same effect takes place in vacuo.”

“Then pray inform us, by what means the top was raised.”

“It entirely depended upon the form of the extremity of the peg, and not upon any simple effect connected with the rotatory or centrifugal force of the top. I will first satisfy you that, were the peg to terminate in a fine, that is to say, in a mathematical point, the top never could raise itself. Let A B C be a top spinning in an oblique position, having the end of the peg, on which it spins, brought to a fine point. It will continue to spin in the direction in which it reaches the ground, without the least tendency to rise into a more vertical position; and it is by its rotatory or centrifugal force that it is kept in this original position: for if we conceive the top divided into two equal parts A and B, by a plane passing through the line XC, and suppose that at any moment during its spinning, the connection between these two parts were suddenly dissolved, then would any point in the part A fly off with the given force in the direction of the tangent, and any corresponding point in the part B with an equal force in an opposite direction; whilst, therefore, these two parts remain connected together, during the spinning of the top, these two equal and opposite forces A and B will balance each other, and the top will continue to spin on its original axis. Having thus shown that the rotatory or centrifugal force can never make the top rise from an oblique to a vertical position, I shall proceed to explain the true cause of this change, and I trust you will be satisfied that it depends upon the bluntness of the point.

Let A B C be a top spinning in an oblique position, terminating in a very short point with a hemispherical shoulder PaM. It is evident that, in this case, the top will not spin upon a the end of the true axis Xa, but upon P, a point in the circle PM, to which the floor IF is a tangent. Instead, therefore, of revolving upon a fixed and stationary point, the top will roll round upon the small circle PM on its blunt point, with very considerable friction, the force of which may be represented by a line OP at right angles to the floor IF, and to the spherical end of the peg of the top: now it is the action of this force, by its pressure on one side of the blunt point of the top, which causes it to rise in a vertical direction. Produce the line OP till it meets the axis C; from the point C draw the line Ct perpendicular to the axis a X, and TO parallel to it; and then, by a resolution of forces, the line TC will represent that part of the friction which presses at right angles to the axis, so as gradually to raise it in a vertical position; in which operation the circle PM gradually diminishes by the approach of the point P to a, as the axis becomes more perpendicular, and vanishes when the point P coincides with the point a, that is to say, when the top has arrived at its vertical position, where it will continue to sleep, without much friction, or any other disturbing force, until its rotatory motion fails, and its side is brought to the earth by the force of gravity.”

“I think I understand it,” said Tom, “although I have some doubt about it; but if you would be so kind as to give me the demonstration in writing, I will diligently study it.”

“Most readily,” said Mr. Seymour. “Indeed I cannot expect that you should comprehend so difficult a subject, without the most patient investigation; and, in the present state of your knowledge, I am compelled to omit the relation of several very important circumstances, to which I may, hereafter, direct your attention. When, for instance, you have become acquainted with the elements of astronomy, I shall be able to show you that the gyration of the top depends upon the same principles as the precession of the equinoxes.”(22)